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Conditions of a Polynomial

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Question

Find the polynomial equation whose roots are the reciprocals of the roots $x^{4} - 3 x^{3} + 7 x^{2} + 5 x - 2 = 0$

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Solution

put $x = \frac{1}{x}$ in the given equation
we get $\frac{1}{x^{4}} - \frac{3}{x^{3}} + \frac{7}{x^{2}} + \frac{5}{x} - 2 = 0$
Therefore required equation is $2 x^{4} - 5 x^{3} - 7 x^{2} + 3 x - 1 = 0$

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